# Calculating heat removed by cooling system using output fluid temperature

I am trying to calculate the amount of energy removed by a cooling system for some medical research. I'm a little out of my depth with the physics calculations.

I have fluid flowing through a variable temperature object (an organ being heated) at a known unchanging flow rate. The fluid input temperature is constant. A thermocouple on output side measures the temperature of the fluid leaving the object at high frequency.

So I have the mass of water, the change in temperature of the water (and obviously it's specific heat capacity). I do not have the average temperature of the water at the end of the experiment.

I think I can calculate the amount of energy removed by taking the area under the curve of the time temperature graph but I'm not sure what the units on the x-axis should be. Would it be mass of water?

Would taking the average temperature of the output water be equally accurate given the flow rate is unchanging? The temperature rises and falls several times.

Thanks so much

The heat that is transferred from the organ to the water can be calculated as the difference between the enthalapy of the inlet and outlet water. Within time $$dt$$ this heat is $$dQ = \dot M C_p (T_\text{out} - T_\text{in}) dt$$ where $$\dot M$$ is the flow rate of the cooling water and $$C_P$$ is its heat capacity. We need to integrate with respect to time: $$Q = \int_0^t \dot M C_p (T_\text{out} - T_\text{in}) dt$$ Only $$T_\text{out}$$ is a function of time: $$Q = \dot M C_p \int_0^t T_\text{out} dt - \dot M C_p T_\text{in} t$$ Therefore we have to calculate the integral $$\int_0^t T_\text{out} dt$$ where $$T_\text{out} = T_\text{out}(t)$$ is the outlet temperature as a function of time.

The following variables apply:

$$Q$$ = the amount of heat transferred out of the organ, BTU/s

$$m$$ = the mass flow rate of the water, lb/s

$$C_p$$ = the specific heat of water, BTU/lb-degF

$$T_i$$ = the temperature of the water entering the organ, deg F

$$T_f$$ = the temperature of the water leaving the organ, deg F

The heat transfer out of the organ over a short time interval can be calculated by the equation $$Q=mC_p(T_f-T_i)$$. By summing up all of the short term heat transfers, you can track the total heat transfer as a function of time.

Note that this method does not require the area under a plot, it does not require the average $$T_f$$, and it does not require higher level math (e.g., integrals) if you record $$T_f$$ at a high frequency (e.g., one or a very few seconds per measurement). In addition, if you are using metric units, the same equation will apply but all of the units will be different.